Below is a photo of my test setup, to verify that the grounding plate as described above actually works:
Here's what's going on in the image: The yellow jumper on the left is for the LED output (in fact, the board here is running the full circuit, so all pins from 12 to 6 are output LEDs; here, it's connected to 11 to demonstrate one of these). In particular, a 100 Ω resistor is connected to pin 11, which then connects to the positive lead on the LED; the negative lead (which I've bent, to remind me which one is which) is then connected directly to ground. A black clip goes from the LED's negative lead (which is equivalent to ground) and goes to the ground plate in the back. Pins 5 and 4 are the output and input pins on the capacitive switch, respectively; pin 5 has a 10 MΩ resistor, then the yellow jumper on the left, which connects to the capacitive plate on the left. Pin 4 has a piece of green wire connected to the rightmost black jumper, which then connects to the capacitive plate right next to pin 5's jumper.
Note that the entirely assembly is located on a glass-topped table; because it's connected to a 9V battery, there is no other grounding. I tested this circuit by holding one hand against the grounding plate and waving my other hand near the capacitive plate (generally, within 0.5 in); this activated the switch. This worked even if I was sitting in a wooden chair without any part of me touching the ground. Thus, by using a second grounding plate, it is possible to build a capacitive switch even in the absence of a true ground.
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